
In Minkowski space, it is known (see [1]}, [2]} and appendix ) that the vanishing of the Cotton tensor associated with the field \(h_{\alpha (n)}\) , is a necessary and sufficient condition for \(h_{\alpha (n)}\)  to be pure gauge. As argued in appendix , an analogous statement holds in AdS space:
\(\mathfrak {C}_{\alpha (n)}(h)=0 \qquad \Longleftrightarrow \qquad h_{\alpha (n)}=\mathcal {D}_{\alpha (2)}\xi _{\alpha (n-2)}~, \) 
